Tunnelling effects for acoustic waves in slowly varying axisymmetric flow ducts
The multiple-scales Wentzel–Kramers–Brillouin (WKB) approximation is used to model the propagation of acoustic waves in an axisymmetric duct with a constriction in the presence of mean flow. An analysis of the reflection/transmission process of modes tunnelling through the constriction is conducted, and the key mathematical feature is the presence of two turning points, located at either real axial locations or in the complex plane. The resulting asymptotic solution consists of WKB solutions in regions away from the constriction and an inner solution valid in the near vicinity of the constriction. A solution which is uniformly valid throughout the duct is also derived. A range of test cases are considered, and the importance of accounting for the inner region, even in cases in which the turning points lie away from the real axis, is demonstrated.